| L(s) = 1 | + 2·4-s − 8·7-s + 16-s − 12·19-s + 8·25-s − 16·28-s − 24·31-s − 4·37-s + 8·43-s + 12·49-s + 72·61-s − 2·64-s + 16·67-s − 24·76-s − 32·79-s + 16·100-s + 96·103-s + 56·109-s − 8·112-s + 2·121-s − 48·124-s + 127-s + 131-s + 96·133-s + 137-s + 139-s − 8·148-s + ⋯ |
| L(s) = 1 | + 4-s − 3.02·7-s + 1/4·16-s − 2.75·19-s + 8/5·25-s − 3.02·28-s − 4.31·31-s − 0.657·37-s + 1.21·43-s + 12/7·49-s + 9.21·61-s − 1/4·64-s + 1.95·67-s − 2.75·76-s − 3.60·79-s + 8/5·100-s + 9.45·103-s + 5.36·109-s − 0.755·112-s + 2/11·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.010576142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.010576142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 38 T^{2} + 673 T^{4} - 10906 T^{6} - 449276 T^{8} - 10906 p^{2} T^{10} + 673 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 12 T + 98 T^{2} + 600 T^{3} + 3027 T^{4} + 600 p T^{5} + 98 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 34 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 92 T^{2} + 4326 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - T + p T^{2} )^{8} \) |
| 47 | \( 1 - 122 T^{2} + 7393 T^{4} - 374906 T^{6} + 18356644 T^{8} - 374906 p^{2} T^{10} + 7393 p^{4} T^{12} - 122 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 61 | \( ( 1 - 18 T + 169 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8 T - 14 T^{2} + 448 T^{3} - 2693 T^{4} + 448 p T^{5} - 14 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 230 T^{2} + 22659 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 122 T^{2} + 9555 T^{4} + 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 104 T^{2} + 13890 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 320 T^{2} + 61246 T^{4} - 8099840 T^{6} + 819799075 T^{8} - 8099840 p^{2} T^{10} + 61246 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 334 T^{2} + 46419 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.99424986439637633818664415403, −3.88605658658334053229738337829, −3.58905243063762032855672901465, −3.55567173492841204591039381589, −3.48574367099021206298892565246, −3.40045522065744235133668477933, −3.29817865962143968661886284977, −3.29122795247649138858690488967, −3.15028136939763680781396950707, −3.11297558409585762684898284266, −2.65064382248941189652197143782, −2.57299995655968140381479285024, −2.30680172020370092263422997709, −2.22470324613043723149188419874, −2.13940576002330532491381338527, −2.11169663435721733875371032799, −2.04952519849642281858244498596, −1.84771921106705036753927978461, −1.71383840539576619857320470764, −1.39137989403283734565056790995, −0.981939641799883627836492430007, −0.71355133495626385748004198988, −0.55859990997517537008671935097, −0.45492873854361321779354912078, −0.44466646345711225123551437892,
0.44466646345711225123551437892, 0.45492873854361321779354912078, 0.55859990997517537008671935097, 0.71355133495626385748004198988, 0.981939641799883627836492430007, 1.39137989403283734565056790995, 1.71383840539576619857320470764, 1.84771921106705036753927978461, 2.04952519849642281858244498596, 2.11169663435721733875371032799, 2.13940576002330532491381338527, 2.22470324613043723149188419874, 2.30680172020370092263422997709, 2.57299995655968140381479285024, 2.65064382248941189652197143782, 3.11297558409585762684898284266, 3.15028136939763680781396950707, 3.29122795247649138858690488967, 3.29817865962143968661886284977, 3.40045522065744235133668477933, 3.48574367099021206298892565246, 3.55567173492841204591039381589, 3.58905243063762032855672901465, 3.88605658658334053229738337829, 3.99424986439637633818664415403
Plot not available for L-functions of degree greater than 10.
